Expected Posterior Prior Distributions for Model Selection

Consider the problem of comparing parametric models M 1 ; : : : ; M k , when at least one of the models has an improper prior N i (i). Using the Bayes factor for comparing among these is not feasible due to arbitrary multiplicative constants in N (i). In this work we suggest adjusting the initial priors for each model, N i , by i (i) = Z N i (i jy)m (y)dy where m is a suitable predictive measure on (imaginary) training samples, y. The updated prior, , is called the expected posterior prior under m. Some properties of this approach include: (1) The resulting Bayes factors depend only on suucient statistics. (2) The resulting Bayesian inference is coherent and allows for multiple comparisons. (3) In many cases, it is possible to nd m such that, for a sample of minimal size, there is predictive matching for the comparisons of model M i to M j ,i.e., the Bayes factor B ij = 1. (4) In the case of nested models, where M 1 is nested in every other model, choosing m (y) to be the marginal of y under M 1 is asymptotically equivalent to the arithmetic Intrinsic Bayes Factor (Berger and Pericchi, 1996). The expected posterior prior scheme can be applied to a wide variety of statistical problems. Applications to the selection of linear models and for the default analysis of mixture models can be seen in P erez (1998).